The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions Hη±(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of . These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ↦ − − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum . Then, it is shown that, for integer or half-integer , the irregular functions Hη±(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,(ρ) and Φη,−−1(ρ) with respect to . As a consequence, the connection formulas directly lead to the description of the singular structures of Hη±(ρ) and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1.

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